A Column Generation Approach for Weighted Set Packing Problems with Forbidden Pairs

The weighted set packing problem (WSPP) is a variant of the well-known set packing problem, assigning a weight to every considered subset of a given finite base set. Its solution is a maximum sum-weight, element-disjoint covering of the base set. Using its formulation as a binary linear program (BLP), every column of the constraint matrix describes one of the given subsets. Thus, column generation (CG) provides a good solution method, generating iteratively only promising subsets as candidates for the covering. We consider an extension of the WSPP, where forbidden pairs are taken into consideration, i.e., there can be pairs of subsets that cannot be chosen at the same time, and adapt CG to derive a new solution algorithm to solve the resulting BLP. At each iteration, the optimal solution obtained with CG is checked for potential conflicts and either they are resolved locally without changing the objective function value or a new linear constraint forbidding the conflicting pair of columns is added. Then the CG procedure starts again. The applicability of this approach is shown for a team orienteering problem involving two unmanned aerial vehicles that must maintain safety distances between them.